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Hypercomputation and the Physical ChurchTuring Thesis
, 2003
"... A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing ..."
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A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, nonwellfounded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard ChurchTuring Thesis.
Can Newtonian systems, bounded in space, time, mass and energy compute all functions?
"... In the theoretical analysis of the physical basis of computation there is a great deal of confusion and controversy (e.g., on the existence of hypercomputers). First, we present a methodology for making a theoretical analysis of computation by physical systems. We focus on the construction and anal ..."
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Cited by 13 (4 self)
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In the theoretical analysis of the physical basis of computation there is a great deal of confusion and controversy (e.g., on the existence of hypercomputers). First, we present a methodology for making a theoretical analysis of computation by physical systems. We focus on the construction and analysis of simple examples that are models of simple subtheories of physical theories. Then we illustrate the methodology, by presenting a simple example for Newtonian Kinematics, and a critique that leads to a substantial extension of the methodology. The example proves that for any set A of natural numbers there exists a 3dimensional Newtonian kinematic system MA, with an infinite family of particles Pn whose total mass is bounded, and whose observable behaviour can decide whether or not n ∈ A for all n ∈ N in constant time. In particular, the example implies that simple Newtonian kinematic systems that are bounded in space, time, mass and energy can compute all possible sets and functions on discrete data. The system is a form of marble run and is a model of a small fragment of Newtonian Kinematics. Next, we use the example to extend the methodology. The marble run shows that a formal theory for computation by physical systems needs strong conditions on the notion of experimental procedure and, specifically, on methods for the construction of equipment. We propose to extend the methodology by defining languages to express experimental procedures and the construction of equipment. We conjecture that the functions computed by experimental computation in Newtonian Kinematics are “equivalent” to those computed by algorithms, i.e. the partial computable functions.
AXIOMATIZING MATHEMATICAL CONCEPTUALISM IN THIRD ORDER ARITHMETIC
"... Abstract. We review the philosophical framework of mathematical conceptualism as an alternative to settheoretic foundations and show how mainstream mathematics can be developed on this basis. The paper includes an explicit axiomatization of the basic principles of conceptualism in a formal system C ..."
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Abstract. We review the philosophical framework of mathematical conceptualism as an alternative to settheoretic foundations and show how mainstream mathematics can be developed on this basis. The paper includes an explicit axiomatization of the basic principles of conceptualism in a formal system CM set in the language of third order arithmetic. This paper is part of a project whose goal is to make a case that mathematics should be disassociated from set theory. The reasons for wanting to do this, which I discuss in greater detail elsewhere ([22]; see also [19] and [23]), involve both the philosophical unsoundness of set theory and its practical irrelevance to mainstream mathematics. Set theory is based on the reification of a collection as a separate object, an elementary philosophical error. Not only is this error obvious, it also has the spectacular consequence of immediately giving rise to the classical set theoretic paradoxes. Of course, these paradoxes are not derivable in the standard axiomatizations of set theory, but that is only because these systems were specifically designed to avoid them. In these systems the paradoxes are blocked by means of ad hoc restrictions on the set concept that have no obvious intuitive justification, which has led to the development of a large literature of attempted rationalizations
Transfinite Machine Models
, 2011
"... In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of co ..."
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In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely. By ‘discrete ’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of course, Turing’s original
A New Problem for Rule Following
"... This is part of an extended argument of mine about the ChurchTuring thesis (CTT). In Hogarth 1994 I argued that the thesis is a thoroughly empirical claim. In Hogarth 2004, 2008 I rejected that view, arguing instead that the thesis is really a pseudoproposition like ‘Australia is below England’, or ..."
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This is part of an extended argument of mine about the ChurchTuring thesis (CTT). In Hogarth 1994 I argued that the thesis is a thoroughly empirical claim. In Hogarth 2004, 2008 I rejected that view, arguing instead that the thesis is really a pseudoproposition like ‘Australia is below England’, or, better, like ‘Euclidean geometry is